Sketch of the Proof of the Second Welfare Theorem, Theorem 5.7*
In this section, I provide a proof of the Second Welfare Theorem. The most important part of the theorem is proved by using the Geometric Hahn-Banach Theorem, Theorem A.25, from Appendix Chapter A
Proof of Theorem 5.7: (Sketch)
First, I establish that there exists a price vector p** and an endowment and share allocation (ω**, θ**^) that satisfy conditions 1-3.
This has two parts.(Part 1) This part follows from the Geometric Hahn-Banach Theorem, Theorem A.25.
P. This implies that if distributed appropriately across the households, y would make all households equally well off and at least one of them would be strictly better off (i.e., by the 201
(Part 2) We next need to show that this linear functional can be interpreted as a price vector (i.e., that it does have an inner product representation). Consider the functional
202
This also implies that
Not taking limits, we obtain that
such that conditions 2 and 3 hold. Condition 1 is satisfied by construction. Condition 2 is
5.8.